The $n$th near-Bell number, as defined by Beck, enumerates all possible partitions of an n-multiset with multiplicities 1,1,1,...,1,2. In this paper we study the sequences arising from a generalization of the near-Bell numbers, and provide a method for obtaining both their exponential and their ordinary generating functions. We derive various interesting relationships amongst both the generating functions and the sequences, and then show how to extend these results to deal with more general multisets.